While it's my understanding that there's no fundamental reason a program written for 32-bit hardware / OSs not to run on 64-bit hardware / OSs, in practice, I've found many programs intended for 32-bit versions of Windows that will not work on 64-bit versions of Windows. Examples include a number of popular security utilities (most products from Norton and Check Point's Zone Alarm) and several games (I've been trying to get Grand Theft Auto 4 to run for a few weeks now, but to no avail - of course, that might be related to any number of other problems related to GTA4, but that's neither here nor there).
I've heard that a program's incompatibility might result from something as simple as not wanting to run from the "Program Files (x86)" folder, but what are some of the other reasons? Why would a virus scanner or firewall written for a 32-bit system not run on a 64-bit system? Why would a game not run when everything is theoretically backwards-compatible?